In modern terms, Euler, de Foncenex, Lagrange, and Laplace were assuming the existence of a splitting field of the polynomial " p " ( " z " ).
32.
Conversely, any Galois extension of " K " of degree " p " equal to the characteristic of " K " is the splitting field of an Artin Schreier polynomial.
33.
The Jordan form can be assumed to exist over a field extending the base field of the matrix, for instance over the splitting field of; this field extension does not change the matrix in any way.
34.
Davesh Maulik, of Roslyn, N . Y ., won fourth place and a $ 15, 000 scholarship for a study of algebraic formulas that he titled " Polynomial Automorphisms of Splitting Fields ."
35.
Ruffini assumed that all radicals that he was dealing with could be expressed from the roots of the polynomial using field operations alone; in modern terms, he assumed that the radicals belonged to the splitting field of the polynomial.
36.
Show that its splitting field can be written in the form L ( \ sqrt { a }, \ sqrt { b } ), where L / K is a Galois cubic extension and a, b " L.
37.
The significance of the Galois group derives from the fundamental theorem of Galois theory, which proves that the fields lying between the ground field and the splitting field are in one-to-one correspondence with the subgroups of the Galois group.
38.
And how do you know that in this case the degree of the splitting field is 24 ( Other than just giving the order of the Galois group of the extension ) ?-- Talk 16 : 50, 16 September 2008 ( UTC)
39.
One can prove this by calculating the dimension of the tensor product over ! as 9, and observing that the splitting field does contain two ( indeed three ) copies of " K ", and is the compositum of two of them.
40.
One of the fundamental theorems of Galois theory states that a polynomial is solvable by radicals over if and only if its splitting field over has a solvable Galois group, so the proof of the Abel Ruffini theorem comes down to computing the Galois group of the general polynomial of the fifth degree.
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